Question

(a) State Faraday’s law of electromagnetic induction and explain the role of the negative sign in its expression.
(b) Explain, with an example, that Lenz’s law is consistent with the law of conservation of energy.

Hint:

For electromagnetic induction problems: - Faraday’s law gives the magnitude of induced EMF: \( \mathcal{E} = -\frac{d\Phi_B}{dt} \). - Lenz’s law (negative sign) ensures the induced current opposes the flux change, aligning with energy conservation.

Answer 1

(a): Faraday’s Law and the Negative Sign.
Faraday’s law of electromagnetic induction quantifies the induced electromotive force (EMF) in a circuit as being directly proportional to the rate of change of magnetic flux through it:\[\mathcal{E} = -\frac{d\Phi_B}{dt},\]where \( \mathcal{E} \) represents the induced EMF, and \( \Phi_B \) denotes the magnetic flux.
The inclusion of the negative sign is a direct consequence of Lenz’s law. This law posits that the direction of the induced current is such that it opposes the very change in magnetic flux that produced it. This opposition is crucial for maintaining the principle of conservation of energy; the induced current must expend energy to counteract the external flux change, thus requiring an input of energy to sustain the process.(b): Lenz’s Law and Energy Conservation.
Lenz’s law dictates that any induced current will flow in a direction that opposes the change in magnetic flux. This fundamental opposition ensures that energy is conserved, as the creation of the induced current necessitates work done against this opposing force.
Example: Imagine a magnet with its north pole moving towards a conductive loop. As the magnet approaches, the magnetic flux through the loop intensifies. According to Lenz’s law, the induced current within the loop generates its own magnetic field, which acts as a north pole facing the approaching north pole of the magnet. This results in a repulsive force, meaning external effort is required to continue moving the magnet closer. The mechanical work performed to overcome this repulsion is transformed into electrical energy within the loop, thereby upholding the law of energy conservation.